Problem 446

Let $\delta(n)$ denote the density of integers which are divisible by some integer in $(n,2n)$ . What is the growth rate of $\delta(n)$ ?

If $\delta_1(n)$ is the density of integers which have exactly one divisor in $(n,2n)$ then is it true that $\delta_1(n)=o(\delta(n))$ ?

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SolvedAnswered

An accepted solution exists. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Solved. The site's label; Ford determined the order of $\delta(n)$ and answered the second question no, as the claim page below records.

Be34: Besicovitch, A., On the density of certain sequences of integers. Math. Annalen (1934), 336-341.
Er35: Erdős, Paul, Note on Sequences of Integers No One of Which is Divisible By Any Other. J. London Math. Soc. (1935), 126-128.
Er60: Erdős, P., An asymptotic inequality in the theory of numbers. Vestnik Leningrad. Univ. (1960), 41-49.
Fo08: Ford, Kevin, The distribution of integers with a divisor in a given interval. Ann. of Math. (2) (2008), 367-433.
Te84: Tenenbaum, G., Sur la probabilité qu'un entier posséde un diviseur dans un intervalle donné. Compositio Math. (1984), 243-263.